Problem Solving

[GMAT math practice question]

There is a string 15 units in length. By cutting the string, Thomas wants to make two equilateral triangles. If the ratio of the areas of two triangles is 2:3, what is the side length of the smaller triangle?

A. 2
B. 3
C. 5( \(\sqrt{6}\) -2)
D. 3( \(\sqrt{5}\) -2)
E. 2 \(\sqrt{6}\)

=>

Assume x and y are the side lengths of the two triangles where x < y.
Then we have 3x + 3y = 15 or x + y = 5.

Since the ratio of the areas of those triangles is 2:3, we have x^2 : y^2 = 2 : 3 or x : y = √2 : √3.
Cross multiplying gives us √3x = √2y, y = (√3x)/√2, and y = (√6/2)x (by multiplying the numerator and denominator by √2).

Substituting y = (√6/2)x into x + y = 5 gives us:
x + y = 5
x + (√6/2)x = 5
(2/2)x + (√6/2)x = 5
√6+2/2x = 5.
Then x = 10/√6+2=10(√6-2)/2=5(√6-2).

Therefore, C is the answer.
Answer: C